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ON THE NUMBER OF POINTS OF NILPOTENT
QUIVER VARIETIES OVER FINITE FIELDS
Tristan Bozec, O. Schiffmann, E. Vasserot
To cite this version:
Tristan Bozec, O. Schiffmann, E. Vasserot. ON THE NUMBER OF POINTS OF NILPOTENT QUIVER VARIETIES OVER FINITE FIELDS. Annales Scientifiques de l’École Normale Supérieure, Société mathématique de France, 2020, �10.24033/asens.2452�. �hal-01709055�
arXiv:1701.01797v1 [math.RT] 7 Jan 2017
VARIETIES OVER FINITE FIELDS
T. BOZEC, O. SCHIFFMANN AND E. VASSEROT
Abstract. We give a closed expression for the number of points over finite fields of the Lusztig nilpotent variety associated to any quiver, in terms of Kac’s A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua’s formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various stratas of the Lusztig nilpotent variety involved in the geometric realization of the crystal graph.
Contents
0. Introduction 2
1. Statement of the result 4
1.1. Lusztig nilpotent quiver varieties 4
1.2. The plethystic exponential 6
1.3. The Kac polynomial 7
1.4. The nilpotent Kac polynomials 7
1.5. Kac polynomials and the nilpotent quiver varieties 8
1.6. Examples 8
1.6.1. Finite Dynkin quivers 9
1.6.2. Affine quivers 9
1.6.3. The Jordan quiver 9
1.7. Remarks 10
1.7.1. Poincar´e duality 10
1.7.2. Kac’s conjecture 10
2. Nilpotent Kac polynomials 11
2.1. Stacks of representations and stacks of pairs 11
2.1.1. From nilpotent endomorphisms to nilpotent Kac polynomials 11
2.1.2. The volume of the stack of nilpotent endomorphisms 12
2.1.3. The volumes of the stack of (1-)nilpotent representations 13
2.2. Hua’s formulas 16
2.2.1. Hua’s formula 16
2.2.2. The 1-nilpotent Hua’s formula 16
2.2.3. The nilpotent Hua’s formula 17
2.3. Proof of Proposition 1.2 18
2.3.1. The integrality 18
2.3.2. The value at 1 18
3. Nakajima’s quiver varieties and their attracting subvarieties 20
3.1. Quiver varieties 20
3.2. Reminder on tori actions on varieties 22
3.2.1. The Bialynicki-Birula decomposition 22
3.2.2. Roots 23
3.3. The Bialynicki-Birula decompositions of M(v, w) 23
3.3.1. The cases of L0(v, w), L(v, w) and M0(v, w). 23
3.3.2. The cases of L1(v, w) and M1(v, w). 25
3.4. More on the Bialynicki-Birula decompositions of M(v, w) 26
3.4.1. The cases of L0(v, w), L(v, w). 26
3.4.2. The cases of L1(v, w). 29
3.4.3. Irreducible components of L0(v, w) and L1(v, w). 29
3.5. Counting polynomials of quiver varieties 30
4. Proof of the theorem 34
4.1. The stratification of L(v, w) 34
4.2. Proof of Theorem 1.4 35
5. Factorization of λQ(q, z) and the strata in Λv. 37
6. Appendix 38
6.1. The generalized quantum group 38
6.2. Character formulas 39
6.3. The generalized Kac-Moody algebra 42
References 44
0. Introduction
The interplay between the geometry of moduli spaces of representations of quivers and the repre-sentation theory of quantum groups has led to numerous constructions and results of fundamental importance for both areas. One of the most fundamental object in the theory is the Lusztig nilpo-tent variety introduced in [20], which is a closed substack Λd of the cotangent stack T∗Rep
d(Q) of
the stack of representations of dimension d of a quiver Q. When Q has no 1-cycle the stack Λd is Lagrangian and as shown by Lusztig (resp. Kashiwara-Saito), its irreducible components are in one to one bijection with the weight d piece of the canonical basis (resp. crystal graph) of U+
q (gQ), where
gQ is the Kac-Moody Lie algebra associated to Q. The stack Λd is singular, and although it can be
inductively built by sequences of (stratified) affine fibrations, see [18], its geometry remains mysterious. The link mentioned above with canonical or crystal bases shows that, for quivers without 1-cycles, the generating series for top Borel-Moore homology groups of Λd is given by
X d dim(Htop(Λd, Q)) zd = X d dim(U+(gQ)[d]) zd = Y α∈∆+ (1 − zα)−dim gQ[α]. (0.1)
A natural problem is to extend the formula (0.1) to the whole cohomology of Λd and to understand its significance from the point of view of representation theory. One first step in this program is worked out in this paper together with its companion [35]. The aim of the present paper is to carry out the essential step in the computation of the cohomology of Λd : we determine the number of points
of Λd over finite fields of large enough characteristic. The answer, given in the form of generating series, is expressed in terms of the Kac polynomials Ad attached to the quiver Q. We refer the reader
to Theorem 1.4 for details. In [35] it is proven that Λd is cohomologically pure, and hence that its Poincar´e polynomial coincides with its counting polynomial. In addition, the (whole) cohomology of Λd is related there to the Lie algebras introduced by Maulik and Okounkov in [23].
Our strategy to compute the number of points of Λd(Fq) is the following : we relate the number of
points of Λd(Fq) to the number of points of certain Lagrangian Nakajima quiver varieties L(d, n)(Fq).
Using some purity result for these Nakajima quiver varieties together with a Poincar´e duality argument we express the counting polynomial of L(d, n) in terms of the Poincar´e polynomial of the symplectic Nakajima quiver variety M(d, n). Finally, we use Hausel’s computation of the Poincar´e polynomials of M(d, n) in terms of Kac polynomials, see [11].
We note that the generating series for the top homology groups of Λd can be extracted from our formula, and involves only the constant terms of Kac polynomials : combining this with (0.1) one recovers a proof of Kac’s conjecture (first proved in [11]) relating the multiplicities of root spaces in Kac-Moody algebras to the constant term of Kac polynomials.
In the context of [23] it is essential to allow for arbitrary quivers Q, such as for instance the quiver with one vertex and g loops. Note that there is no Kac-Moody algebra associated to a quiver which does carry 1-cycles. In [1], [2] the first author introduced a quantum group Uq(gQ) attached to an arbitrary
quiver Q which coincides with the usual quantized Kac-Moody algebra for a quiver with no 1-cycles and, he generalized to this context several fundamental constructions and results, in particular the theory of canonical and crystal bases, and an analogue of (0.1). In the presence of 1-cycles, Lusztig’s nilpotent variety is not Lagrangian anymore and one has to consider instead a larger subvariety Λ1 defined by some ‘semi-nilpotency’ condition. In addition, when the quiver Q contains some oriented cycle we consider yet a third subvariety Λ0 defined by some weak form of semi-nilpotency. The varieties Λ0, Λ1 are Lagrangian, contains Λ and are in some sense more natural than Λ from a geometric perspective. We carry out in parallel the computation of the number of points over finite fields for each of the Λ0, Λ1 and Λ. This leads us to introduce two variants A0 and A1of the Kac polynomials, respectively counting
nilpotent and 1-cycle nilpotent indecomposable representations, see Section 1.4 for more details, for which we prove the existence and give an explicit formula similar to Hua’s formula. As an application, we provide a proof of an extension of Kac’s conjecture on the multiplicities of Kac-Moody Lie algebras to the setting of arbitrary quivers.
To finish, let us briefly describe the contents of this paper : the main actors are introduced and our main theorem is stated in Section 1, where several examples are explicitly worked out. Section 2 deals with the existence of nilpotent Kac polynomials A0, A1, and provides explicit formulas for these in the spirit of Hua’s formula for the usual Kac polynomial. In Section 3 we study several subvarieties L0(v, w), L(v, w), L1(v, w) of the symplectic Nakajima quiver variety M(v, w), respectively corre-sponding to the Lusztig nilpotent varieties Λ0, Λ1 and Λ. More precisely, we establish some purity
argument (based on Byalinicki-Birula decompositions) with Hausel’s computation of the Betti num-bers of M(v, w). In Section 4 we relate the counting polynomials of Λv, Λ0v, Λ1v to the the counting
polynomials of the L(v, w), L0(v, w), L1(v, w) and prove our main theorem. Section 5 contains an ob-servation about the counting polynomials of certain natural strata in Lusztig nilpotent varieties arising in the geometric realization of crystal graphs. Finally, in the appendix we recall the definition of the quantum group associated in [2] to an arbitrary quiver, give a character formula for it, and use our main Theorem to prove an extension of Kac’s conjecture in that context.
1. Statement of the result 1.1. Lusztig nilpotent quiver varieties.
Let Q = (I, Ω) be a finite1 quiver, with vertex set I and edge set H. For h ∈ Ω we will denote by h′, h′′ the initial and terminal vertex of h. Note that we allow 1-loops, i.e., edges h satisfying h′ = h′′.
Set v · v′ =Pivivi′. We denote by
hv, v′i = v · v′−X
h∈Ω
vh′v′
h′′
the Euler form on ZI, and by (•, •) its symmetrized version such that (v, v′) = hv, v′i + hv′, vi. We will call imaginary (resp. real) a vertex which carries a 1-loop (resp. which doesn’t carry an 1-loop) and write I = Iim⊔ Ire for the associated partition of I.
Let Q∗ = (I, Ω∗) be the opposite quiver, in which the direction of every arrow is inverted. Let
¯
Q = (I, ¯Ω) with ¯Ω = Ω ⊔ Ω∗ be the doubled quiver, obtained from Q by replacing each arrow h by a pair of arrows (h, h∗) going in opposite directions.
Fix a field k. For each dimension vector v ∈ NI we fix an I-graded k-vector space V = LiVi of
graded dimension v and we set Ev= M h∈Ω Hom(Vh′, Vh′′), E∗ v= M h∈Ω∗ Hom(Vh′, Vh′′), E¯v= M h∈ ¯Ω Hom(Vh′, Vh′′).
Elements of Ev, Ev∗ and ¯Ev will be denoted by x = (xh), x∗ = (xh∗) and ¯x = (x, x∗).
By a flag of I-graded vector spaces in V we mean a finite increasing flag of I-graded subspaces {0} = L0 (L1 (· · · ( Ls = V. We’ll say that (Ll) is a restricted flag of I-graded vector spaces if for all l the vector space Ll/Ll−1 is concentrated on one vertex. Since we want to consider arbitrary
quivers, we need several notions of nilpotency for quiver representations. We say that : • x is nilpotent if there exists a flag of I-graded vector spaces (Ll) in V such that
xh(Ll) ⊆ Ll−1, l = 1, . . . , s, h ∈ Ω,
• x is 1-nilpotent if for any vertex i ∈ Iim there exists a flag of subspaces (Ll
i) in Vi such that
xh(Lli) ⊆ Ll−1i l = 1, . . . , s, h ∈ Ω with h′ = h′′= i, 1locally finite quiver would do as well
• ¯x is nilpotent if there exists a flag of I-graded vector spaces (Ll) in V such that
xh(Ll) ⊆ Ll−1, xh∗(Ll) ⊆ Ll−1, l = 1, . . . , s, h ∈ Ω.
• ¯x is semi-nilpotent2 if there exists a flag of I-graded vector spaces (Ll) in V such that xh(Ll) ⊆ Ll−1, xh∗(Ll) ⊆ Ll, l = 1, . . . , s, h ∈ Ω,
• ¯x is strongly semi-nilpotent if there is a restricted flag of I-graded vector spaces (Ll) in V with xh(Ll) ⊆ Ll−1, xh∗(Ll) ⊆ Ll, l = 1, . . . , s, h ∈ Ω,
• switching the roles of Ω and Ω∗ we get the notion of ∗-nilpotent and ∗-strongly
semi-nilpotent representation.
The sets of nilpotent and 1-nilpotent representations in Ev form Zariski closed subvarieties Ev0, E1v
of Ev. Likewise, the set of nilpotent, strongly nilpotent, ∗-nilpotent, ∗-strongly
semi-nilpotent and semi-nilpotent representations in ¯Ev form closed subvarieties ¯Nv, ¯Nv1, ¯Nv∗, ¯N ∗,1
v and ¯Ev0 of ¯Ev
respectively. We have the following inclusions
Ev0 ⊆ Ev1 ⊆ Ev, E¯v0 ⊆ ¯Nv1⊆ ¯Nv⊆ ¯Ev, E¯v0 ⊆ ¯Nv∗,1 ⊆ ¯Nv∗ ⊆ ¯Ev.
Remark 1.1. (a) For any path σ in Q, let xσ be the composition of x along σ. The representation x
of Q is nilpotent if and only if there is an integer N such that xσ = 0 for each path σ of length > N .
Similarly, the representation ¯x of ¯Q is nilpotent if and only if ¯xσ = 0 for each path σ of length > N
for some N . Further, ¯x is semi-nilpotent if there exists N such that ¯xσ = 0 for each path σ containing
at least N arrows in Q, see Proposition 3.1(c).
(b) Assume that Q has no oriented cycle which does not involve 1-loops, i.e., any oriented cycle in Q is a product of 1-loops. Then a representation x of Q is nilpotent if and only if it is 1-nilpotent, i.e., we have Ev0 = Ev1. In addition, the definitions of semi-nilpotent and strongly semi-nilpotent representations coincide, i.e., we have ¯Nv1= ¯Nv and likewise for Q∗.
(c) Assume now that Q contains no 1-loop. Then all representations are 1-nilpotent, i.e., we have Ev1 = Ev, and any strongly semi-nilpotent representation is automatically nilpotent, i.e., we have
¯
Ev0 = ¯Nv1 = ¯Nv∗,1.
(d) Let us finally assume that Q has no oriented cycle at all. In that case, we have Ev0 = Ev1 = Evand
all the various semi or ∗-semi-nilpotency conditions for elements in ¯Evare equivalent to the nilpotency
condition, i.e., we have ¯E0v= ¯Nv= ¯Nv1= ¯Nv∗= ¯N ∗,1 v .
In the presence of 1-loops or oriented cycles however the above definitions of semi-nilpotency and nilpotency do differ and we only have ¯Ev0 ⊂ ¯Nv and ¯Ev0 ⊆ ¯Nv∗. This is already the case for the Jordan
quiver.
The group Gv =QiGL(Vi) acts on Ev, ¯Ev by conjugation. This action preserves the subsets ¯Nv,
¯
Nv∗ and ¯Ev0.
2Our definition of semi-nilpotent representation is not the same as that appearing in [2]; what was called semi-nilpotent
The trace map Tr : ¯Ev → k such that ¯x 7→ Ph∈ΩTr(xhxh∗) identifies ¯Ev with T∗Ev. We set
gv = Lie(Gv) = Ligl(Vi) and identify gv with its dual g∗v via the trace. The moment map for the
action of Gv on ¯Ev= T∗Ev is the map µv: ¯Ev→ gv such that ¯x 7→Ph∈Ω[xh, xh∗].
Following Lusztig [20], we consider the variety
Λv = µ−1v (0) ∩ ¯Ev0,
which is often called the Lusztig nilpotent variety. It is a closed subvariety of ¯Ev which, in general,
possesses many irreducible components and is singular. When Q has no 1-loop Λv is Lagrangian, but
in general it is not of pure dimension. Following [2], we also set
Λ1v= µ−1v (0) ∩ ¯Nv1, Λ∗,1v = µ−1v (0) ∩ ¯Nv∗,1,
which are both Lagrangian subvarieties in ¯Ev. The above varieties play an important role in the
geometric approach to quantum groups and crystal graphs based on quiver varieties. We’ll call them the strongly semi-nilpotent varieties. See [18] or [33, lect. 4] and [2] for the case of quiver with 1-loops. Finally, we set
Λ0v= µ−1v (0) ∩ ¯Nv, Λ∗,0v = µ−1v (0) ∩ ¯Nv∗.
These are again Lagrangian subvarieties in ¯Ev. These varieties have received less attention than the
previous ones because they are not directly linked to quantum groups, but they are very natural from a geometric point of view. We’ll call them the semi-nilpotent varieties. We have
Λv⊆ Λ1v⊆ Λ0v, Λv⊆ Λ∗,1v ⊆ Λ∗,0v .
When the quiver Q has no 1-loop, we have Λv = Λ1v = Λ ∗,1
v . When the quiver has no oriented cycle
besides the 1-loops we have Λ1v= Λ0v and Λ∗,1v = Λ∗,0v .
The map µ and hence the varieties Λv, Λ♭v, Λ ∗,♭
v with ♭ = 0, 1 are defined over an arbitrary field k.
The aim of this paper is to establish a formula for the number of points of all these varieties over finite fields, in terms of Kac’s A-polynomials [16]. As usual, let q be a power of a prime number p. We will give formulas for the generating series of |Λv(Fq)|, |Λ1v(Fq)| and |Λ0v(Fq)|.
1.2. The plethystic exponential.
Before we can state our result, we need to fix a few notations. Consider the spaces of power series L= Q[[zi; i ∈ I]], Lt= Q(t)[[zi; i ∈ I]].
Here t and zi are formal variables. For v ∈ NI we write zv=Qizivi. Let
Exp : Lt→ Lt
be the plethystic exponential map, which is given by Exp(f ) = exp X
l>1
ψl(f )/l
, where ψl: Lt→ Lt is the lth Adams operator defined by
1.3. The Kac polynomial.
For v a dimension vector of Q let Av(t) be the Kac polynomial attached to Q and v. For any
finite field Fq, the integer Av(q) is equal to the number Av(Fq) of isomorphism classes of absolutely
indecomposable representation of Q of dimension v over Fq. The existence of Av(t) is due to Kac and
Stanley, see [16], as is the fact that Av(t) ∈ Z[t] is unitary of degree 1 − hv, vi. That Av(t) has positive
coefficients was only recently proved in [13]. By Kac’s theorem, we have Av(t) = 0 unless v belongs to
the set ∆+ of positive roots of Q. Let us consider the formal series in L
tgiven by PQ(t, z) = Exp 1 1 − t−1 X v Av(t−1) zv . (1.1)
Write Av(t) =Pnav,ntn. The definition of PQ(t) may be rewritten as follows :
PQ(t, z) = Y v∈∆+ Y l>0 1−hv,viY n=0 (1 − t−n−lzv)−av,n.
Observe that the Fourier modes of PQ(t, z) are all rational functions in t regular outside of t = 1. This
allows us to evaluate PQ(t, z) at any t 6= 1.
1.4. The nilpotent Kac polynomials.
To express the point count of Λ♭v with ♭ = 0, 1 we need some variants of the Kac polynomial. For any finite field Fq we let A0v(Fq) and A1v(Fq) be the number of absolutely indecomposable nilpotent
and 1-nilpotent representations of Q of dimension v. By construction, we have Av(Fq) > A1v(Fq) > A0v(Fq).
We establish in §§2, 4 the following
Proposition 1.2. For any quiver Q and any dimension v ∈ NI there are unique polynomials A♭ v(t) in
Z[t] such that for any finite field Fq we have A♭v(q) = A♭v(Fq). Moreover, we have A♭v(1) = Av(1).
We define the formal series in Ltgiven by
PQ♭(t, z) = Exp 1 1 − t−1
X
v
A♭v(t−1) zv. (1.2)
Remark 1.3. (a) We conjecture that A♭
v(t) ∈ N[t].
(b) If Q has no 1-loops then A1v(t) = Av(t).
(c) If any oriented cycle in Q is a product of 1-loops then A1
1.5. Kac polynomials and the nilpotent quiver varieties.
For an arbitrary quiver Q we consider the generating functions in L given by λQ(q, z) = X v |Λv(Fq)| |Gv(Fq)| qhv,vizv, λ♭Q(q, z) =X v |Λ♭v(Fq)| |Gv(Fq)| qhv,vizv. Our main result reads as follows.
Theorem 1.4. If the field Fq is large enough then we have
λ♭Q(q, z) = PQ♭(q, z), λQ(q, z) = PQ(q, z). (1.3)
Remark 1.5. (a) By a large enough field Fq we mean the following : for each dimension vector v ∈ NI
there exists an integer N > 0 such that the equality (1.3) holds in degrees v′ 6v for any field Fq of
characteristic p > N .
(b) We define the formal series
ζv(q, u) = Exp (Av(q) u) = exp X l>1 Av(ql) ul/l . This may be thought of as the ’zeta function’ of the quotient
Ev(Fq) = {x ∈ Ev(Fq) ; (xh⊗ Fq) indecomposable}/Gv.
The set Ev(Fq) is not the set of Fq-points of an algebraic variety. It is only the set of Fq-points of a
constructible subset of the moduli stack Ev/Gv of representations of Q of dimension v. So we can not
speak of its zeta function. Using this notation, we may restate (1.3) as the equality λQ(q, z) = Y v∈∆+ Y l>0 ζv(q−1, q−lzv).
A similar interpretation may be given for λ♭Q(z, q) in terms of the ’zeta functions’ of stacks of 1-nilpotent, resp. nilpotent, indecomposable representations.
(c) It is possible to give a closed expression for the power series PQ(t, z) and PQ♭(t, z), as an immediate
consequence of Hua’s formula and their nilpotent variants established in §2, see (2.8), (2.10) and (2.12). (d) The formulas for the point count of Λ♭(v) in Theorem 1.4 also hold in the Grothendieck group of varieties (or more precisely stacks, see [3]) over an algebraically closed field. One simply replaces every occurence of q by the Lefschetz motive L. This is a consequence of the fact that the point count of the Nakajima quiver varieties performed in [11] admit a similar motivic lift, see [37].
1.6. Examples.
1.6.1. Finite Dynkin quivers. Assume that Q is a finite Dynkin quiver. Then, we have Av(t) = A♭v(t) =
1 for all v ∈ ∆+, and thus
λQ(q, z) = λ♭Q(q, z) = Y v∈∆+ Y l>0 (1 − q−lzv)−1 = Exp q q − 1 X v∈∆+ zv.
1.6.2. Affine quivers. Assume that Q is an affine quiver. Then ∆+= ∆+im⊔ ∆+re with
∆+im= N>1δ, ∆+re= {∆+0 + Nδ} ⊔ {∆−0 + N>1δ}
where δ is the minimal positive imaginary root, and ∆0 is the root system of an underlying finite type
subquiver Q0 ⊂ Q. We have
Av(t) = A♭v(t) = 1 for v ∈ ∆+re
while setting r = rank(Q0) = |I| − 1 we have
Av(t) = A♭v(t) = t + r for v ∈ ∆+im
when Q is not a cyclic quiver and
Av(t) = A1v(t) = t + r, A0v(t) = r + 1 for v ∈ ∆+im
if Q is a cyclic quiver. This follows from the explicit classification of indecomposable representations of euclidean quivers, see [4]. This yields
λQ(q, z) = λ1Q(q, z) = Exp P v∈∆+0 q zv+ (1 + rq) zδ+ P v∈∆− 0 q z v+δ (q − 1)(1 − zδ) ! , λ0Q(q, z) = Exp P v∈∆+0 q zv+ (1 + r)q zδ+ P v∈∆− 0 q z v+δ (q − 1)(1 − zδ) !
for the cyclic quiver of type A(1)r , and
λ♭Q(q, z) = λQ(q, z) = Exp P v∈∆+0 q zv+ (1 + rq) zδ+ P v∈∆− 0 q z v+δ (q − 1)(1 − zδ) !
for all other affine quivers.
1.6.3. The Jordan quiver. Assume that Q is the Jordan quiver (with one vertex and one loop). Then ∆+= N>1 and we have
Av(t) = t, A♭v(t) = 1 for all v > 1.
Further Λ♭vis the variety of pairs of commuting v × v-matrices, with the second matrix being nilpotent while Λv is the variety of pairs of commuting nilpotent matrices. Theorem 1.4 gives
λ♭Q(q, z) = Exp qz (q − 1)(1 − z) = Y v>1 Y l>0 (1 − q−lzv)−1, λQ(q, z) = Exp z (q − 1)(1 − z) = Y v>1 Y l>1 (1 − q−lzv)−1.
The above formula should be compared to the Feit-Fine formula for the number of points in the (non-nilpotent) commuting varitieties over finite fields, see [8]. Similar formulas also appear in [31]. 1.7. Remarks.
We finish this section with several short remarks. 1.7.1. Poincar´e duality. It is not difficult to show that
X v |µ−1 v (0)(Fq)| |Gv(Fq)| qhv,vizv = Exp q q − 1 X v Av(q) zv ! . (1.4)
See [25, thm. 5.1] for a similar result in the context of Donaldson-Thomas theory. Comparing (1.4) with (1.1) we see that, as far as point counting goes, the quotient stacks [Λv/Gv] and [µ−1v (0)/Gv] are
in some sense Poincar´e dual to each other. In fact, our proof of Theorem 1.4 uses such a Poincar´e duality for Nakajima varieties, which are framed, stable versions of Λv and µ−1v (0), see §2.1.
1.7.2. Kac’s conjecture. Let Q be a quiver without 1-loops and let g be the associated Kac-Moody algebra. By the theorem of Kashiwara-Saito [18], conjectured by Lusztig in [21], the number of irre-ducible components of Λv = Λ1v is the dimension of the v weight space U+(g)[v] of the envelopping
algebra U+(g). By the Lang-Weil theorem, we have
|Λv(Fq)| = |Irr(Λv)| qdim(Λv)+ O(qdim(Λv)−1/2)
from which it follows that |Λv(Fq)|
|Gv(Fq)|
qhv,vi = |Irr(Λv)| + O(q−1/2) = dim(U+(g)[v]) + O(q−1/2)
which we may write as
λQ(q, z) = X v dim(U+(g)[v]) zv+ O(q−1/2), =Y v (1 − zv)−dim(g[v])+ O(q−1/2). (1.5)
On the other hand, by Theorem 1.4 we have λQ(q, z) = Exp X v av,0zv + O(q−1/2) =Y v (1 − zv)−av,0 + O(q−1/2). (1.6)
Combining (1.5) and (1.6) we obtain that av,0 = dim(g[v]), which is the statement of Kac’s conjecture.
This conjecture was proved by Hausel in [11], using his computation of the Betti numbers of Nakajima quiver varieties. Note that our derivation of Theorem 1.4 uses Hausel’s result in a crucial manner, so that the above is not a new proof of Kac’s conjecture but rather a reformulation of Hausel’s proof in terms of Lusztig nilpotent varieties instead of Lagrangian Nakajima quiver varieties (more in the spirit of [6]).
We refer to the appendix for a proof of an extension of Kac’s conjecture to the setting of an arbitrary quiver. This now involves the Lie algebras introduced in [2].
2. Nilpotent Kac polynomials
Given a category C, let C be the groupoid formed by the objects of C with their isomorphisms. If C is finite, we denote by vol(C) the (orbifold volume)
vol(C) =X
M
1 |Aut(M )|,
where the sum ranges over all isomorphism classes of objects M in C and |Aut(M )| is the number of elements in Aut(M ). Given an algebraic group G acting on a variety X, let X//G be the categorical quotient and [X/G] be the quotient stack.
2.1. Stacks of representations and stacks of pairs.
Let us consider an arbitrary quiver Q. Denote by gi be the number of 1-loops at the vertex i ∈ I.
The aim of this section is to prove the existence of polynomials A0v(t), A1v(t) ∈ Z[t] counting nilpotent and 1-nilpotent, absolutely indecomposable representations of Q of dimension v, and to give an explicit formula for them in the spirit of Hua’s formula, see [15].
2.1.1. From nilpotent endomorphisms to nilpotent Kac polynomials. Let Ck be a Serre subcategory of
the category Repk of all representations of Q over k. It is an abelian category of global dimension one.
Let Cv,k be the stack of all representations of Q of dimension v over k which belong Ck. Let Cnilv,k be
the stack of pairs (M, θ) with M ∈ Cv,k and θ ∈ End(M ) a nilpotent endomorphism.
Now, assume that k is a finite field. For each dimension vector v we consider the volume of Cv,k is vol(Cnilv, k) = X
(M,θ)
1 |Aut(M, θ)|.
Finally, let A(Cv,k) be the number of absolutely indecomposable representations in Cv, k.
Proposition 2.1. The following relation holds in L X
v
vol(Cnilv, Fq) zv= exp X
v X l>1 A(Cv, Fql) ql− 1 z lv/l. (2.1)
Proof. The proof follows closely that of [34, prop. 2.2] and is essentially based on the Krull-Schmitt property of CFq together with the fact that objects in CFq have connected automorphism groups. We
briefly sketch the argument for the comfort of the reader and refer to loc. cit. for details. Let M ∈ CFq and let M =
L
iM ⊕ni
i be a decomposition of M as a direct sum of indecomposables.
Note that the (Mi, ni) are uniquely determined up to a permutation. By the Wedderburns’ theorem,
the ring End(Mi)/ rad(End(Mi)) is a field extension of Fq. We set
Then, we have rad(End(M )) =M i6=j Hom(M⊕ni i , M ⊕nj j ) ⊕ M i rad(End(Mi))⊕ni, Aut(M ) =Y i Aut(M⊕ni i ) ⊕ M i6=j Hom(M⊕ni i , M ⊕nj j ), End0(M ) =Y i End0(M⊕ni i ) ⊕ M i6=j Hom(M⊕ni i , M ⊕nj j ). (2.2)
Let End0 denote the set of nilpotent endomorphisms. We deduce | End0(M )| | Aut(M )| = Y i | End0(M⊕ni i )| | Aut(M⊕ni i )| =Y i qlini(ni−1) |GL(ni, Fqli)| . (2.3)
Let Ind(Ck) be the set of all isomorphism classes of indecomposable objects in Ck, and let us choose
representatives Mi for each i ∈ Ind(Ck). Summing (2.3) over all objects M yields an equality
X v vol(Cnilv, Fq) zv = Y i∈Ind X n>0 q−lin (1 − q−lin) · · · (1 − q−li)z n dim(Mi) . Applying Heine’s formula
X n>0 un (1 − vn) · · · (1 − v) = exp X l>1 ul l(1 − vl) we get X v
vol(Cnilv, Fq) zv= exp X
l>1 X i∈Ind(CFq) zldim(Mi) l(qlli− 1) . To prove Proposition 2.1 it only remains to show the equality
X l>1 X i∈Ind(CFq) zldim(Mi) l(qlli− 1) = X l>1 X v A(Cv, Fql) l(ql− 1) z lv. (2.4)
This last equality follows from a standard Galois cohomology argument, see [34, lem. 2.6]. ⊓⊔ 2.1.2. The volume of the stack of nilpotent endomorphisms. Our aim now is to compute the left hand side of (2.1), and to deduce from this both the existence of and a formula for A(Cv, k). For this we
introduce a stratification of Cnilv, k by Jordan type as follows : to any object (M, θ) in Cnilv, k with θs= 0
and θs−16= 0 we associate the sequence of surjective maps
M/Im(θ) u1 ////Im(θ)/Im(θ2) u2 ////· · · us−1////Im(θs−1) us ////0 .
Then, we define the Jordan type of (M, θ) as
Observe that we havePii vi = v. We obtain a partition
Cnilv, k =G
v
Cnilv, k
where v ranges over all tuples (v1, . . . , vs) such thatPii vi = v and
Cnilv, k = {(M, θ) ∈ Cnilv, k; J(M, θ) = v}.
Let E → F be a map of vector bundles on an algebraic stack. Then the quotient stack of E by F is an algebraic stack. We call such a stack a stack vector bundle. We define
o(v) = −X i (i − 1)hvi, vii − X i<j i (vi, vj). (2.5)
The following result is proved just as in [34, prop 3.1], see also [26, §5]. Lemma 2.2. The morphism πv : Cnilv, k →
Q
iCvik given by
πv(M, θ) = (Ker(u1), Ker(u2), . . . , Ker(us))
is a stack vector bundle of rank o(v). ⊓⊔
We deduce as in [9, §3.1] that
Corollary 2.3. The volume of Cnilv, Fq is given by vol(Cnilv, Fq) =X
v
qo(v)Y
i
vol(Cvi, Fq),
where the sum ranges over all tuples v = (v1, v2, . . . , vs) withPii vi= v. ⊓⊔
2.1.3. The volumes of the stack of (1-)nilpotent representations. Let Rep0k and Rep1kbe the category of nilpotent and 1-nilpotent representations of Q over some field k. By Proposition 2.1 and Corollary 2.3, the computation of A0v(q) and A1v(q) (for all v and all q) reduces to the computation of the volumes vol(Rep0v, F
q) and vol(Rep
1
v,Fq) (for all v and all q).
We begin with vol(Rep1v,F
q). For integers n, g ∈ N we consider the elements [∞, n]t and H(n, g)t in
Q(t) given by [∞, n]t= n Y k=1 (1 − tk)−1 if n > 0, 1 if n = 0, H(n, g)t= X (ek) t(g−1)Pk<lekel+g P k(ek)2Y k [∞, gek− ek+1]t[∞, ek+1]t [∞, gek]t if g > 0, [∞, ni]t if g = 0,
where the sum runs over all tuples (ek) of positive integers with sum Pkek = n. More generally, for n∈ NI we set [∞, n]t= Y i [∞, ni]t, H(n)t= Y i∈I H(ni, gi)t.
Recall that gi is the number of loops at an imaginary vertex i ∈ Iim.
Lemma 2.4. For any v ∈ NI we have
vol(Rep1v, F
q) = q
−hv,viH(v) q−1.
Proof. Set k = Fq. Let Nl(g)be set of g-tuples of nilpotent matrices in End(kl), i.e., the set of 1-nilpotent
representations of dimension l of the quiver with one vertex and g loops. We have Rep1
v, k = [Xv/Gv] where Xv= Y h∈Ω h′ 6=h′′ Hom(kvh′ , kvh′′ ) × Y i∈Iim N(gi) vi .
We now compute the number of elements |Nl(g)| in Nl(g). To a point x = (x1, . . . , xg) in Nl(g) we
associate a flag F = (Fk) of subspaces in kl as follows :
Fk= g
X
h1,...,hk=1
Im(xh1· · · xhk).
We set fk(x) = dim(Fk), and define a partition Nl(g)=FfN (g) f where
Nf(g) = {x ∈ Nl(g); fk(x) = fk, ∀k}.
Given a flag F of dimension f = (fk), we count the number of tuples x whose associated flag is F .
This is a (proper) open subset of (nF)g where nF = {y ∈ End(kl) ; y(Fk) ⊆ Fk+1, ∀k}. The number of
surjective linear maps from ka to kb is equal to
|GLa(k)| / qb(a−b)|GLa−b(k)| = qab
[∞, a − b]q−1
[∞, a]q−1
. (2.6)
We deduce that the number of tuples x associated with F is equal to
qgPk<hekehY
k
[∞, gek− ek+1]q−1
[∞, gek]q−1
where ek = fk−1− fk for all k. Summing over all flags of dimension f = (fk) gives |Nf(g)| = |GLl(k)| q− P k6hekehqgPk<hekehY k [∞, gek− ek+1]q−1[∞, ek+1]q−1 [∞, gek]q−1 = q (g+1)Pk<hekeh [∞, l]q−1 Y k [∞, gek− ek+1]q−1[∞, ek+1]q−1 [∞, gek]q−1 .
Summing over all possible dimensions f yields |Nl(g)|. ⊓⊔
Let us now deal with the similar case of vol(Rep0
v,Fq). For dimension vectors v, w ∈ N
I we set H(v, w, t) =Y i [∞,Ph′′ =ivh′− wi]t [∞,Ph′′=ivh′]t
Lemma 2.5. For any v ∈ NI we have
vol(Rep0 v, Fq) = X v q−a(v)Y l H(v(l), v(l+1), q−1)Y l [∞, v(l)]q−1 where a(v) =X l>k hv(k), v(l)i +X i,l (v(l)i )2
and where the sum ranges over all tuples v = (v(1), v(2), . . .) of elements of NI such thatP
lv(l)= v.
Proof. We stratify the stack Rep0v,F
q as follows. To a nilpotent representation x we associate the
decreasing flag U (x) given by V = U0 ⊃ U1 ⊃ U2 ⊃ · · · Ul(x) ⊃ Ul(x)+1 = {0} where, for each i > 1, we
have Ui= X h xh(Ui−1). We set v(k)(x) = dim(U
k−1/Uk) ∈ NI and for v = (v(1), v(2), . . .) we set
Rep0v,F q[v] = {x ∈ Rep 0 v,Fq ; v (l)(x) = v(l), l > 1} so that Rep0v,F q = G v Rep0v,F q[v]
where the sum ranges over all tuples v = (v(k))
k satisfying Plv(l)= v. Next, we compute the volume
of each strata. Given a flag U• in V of dimension v, the set of nilpotent representations x satisfying
U (x) = U• is isomorphic to M l>k+1 L(Uk/Uk+1, Ul/Ul+1) ⊕ M l Lsurj(Ul/Ul+1, Ul+1/Ul+2).
For each I-graded vector spaces V, W we set L(V, W ) =M
h
Hom(Vh′, Wh′′), Lsurj(V, W ) = {(yh)h ∈ L(V, W ) ; ∀ i ∈ I, Im(
M
h′′
=i
yh) = Wi}.
The volume of the substack of Rep0v,F
q whose objects are the representations x such that U (x) = U• is
thus equal to qPl>k P hv (k) h′ v (l) h′′Y l Y i∈I [∞,Ph′′ =iv (l) h′ − v (l+1) i ]q−1 [∞,Ph′′ =iv (l) h′]q−1 .
It only remains to sum up over all flags U• of dimension v, and to sum up over all possible tuples v. ⊓⊔
2.2. Hua’s formulas.
2.2.1. Hua’s formula. In this section we write down the analogues of Hua’s formula, for our nilpotent versions of Kac polynomials. We begin by recalling the original Hua’s formula (or a formula equivalent to it). Let ν = (νi; i ∈ I) be an I-partition, i.e., an I-tuple of partitions. Consider the I-tuples of integers |ν| and νk in NI given, for each k = 1, 2, . . . , by
|ν| = (|νi|), νk= (νki). Let X(ν, t) ∈ Q(t) be given by X(ν, t) =Y k thνk,νki[∞, ν k− νk+1]t.
We define a power series r(w, t, z) in Lt depending on a tuple w ∈ ZI as
r(w, t, z) =X
ν
X(ν, t−1) tw·ν1z|ν|. (2.7)
Then Hua’s formula [15] is
Exp 1 t − 1 X v Av(t) zv = r(0, t, z). (2.8)
2.2.2. The 1-nilpotent Hua’s formula. Given an I-partition ν as in the previous section, we consider the element X1(ν, t) ∈ Q(t) given by
X1(ν, t) =Y
k
thνk,νki H(ν
k− νk+1)t.
Next, we define a power series in r1(w, t, z) ∈ Lt depending on a vector w ∈ ZI as
r1(w, t, z) =X
ν
Note that if v = (v1, . . . , vs) is a tuple of elements in NI such that νk− νk+1 = vk for all k, and if the
integer o(v) is as in (2.5), then we have o(v) −X k hvk, vki = − X k hνk, νki.
We can now state the following analogue of Hua’s formula, which is a direct consequence of Propo-sition 2.1, Corollary 2.3 and Lemma 2.4.
Corollary 2.6. For any v ∈ NI there exists a unique polynomial A1
v(t) ∈ Q[t] such that for any finite
field Fq we have A1v(q) = A1v(Fq). These polynomials are defined by the following identity :
Exp 1 t − 1
X
v
A1v(t) zv= r1(0, t, z). (2.10)
2.2.3. The nilpotent Hua’s formula. Given a sequence ν• = (ν(1), . . . , ) of I-partition as in the previous section, for each k, l > 1 we set
|ν•| =X l |ν(l)|, νk= X l νk(l), v(l)k = νk(l)− νk+1(l) , and we consider the element X0(ν•, t) in Q(t) given by
X0(ν•, t) = tb(ν•) Y l,k [∞, v(l)k ]tH(v(l)k , v(l+1)k )t where b(ν•) =X k hνk, νki − X l16l2 hv(l1) k , v (l2) k i + X i,l (v(l)k )2. Next, we define a power series in r0(w, t, z) in Lt depending on a vector w ∈ ZI as
r0(w, t, z) =X
ν•
X0(ν•, t−1) tw·ν1z|ν•|. (2.11)
Corollary 2.7. For any v ∈ NI there exists a unique polynomial A0v(t) ∈ Q[t] such that for any finite field Fq we have A0v(q) = A0v(Fq). These polynomials are defined by the following identity :
Exp 1 t − 1
X
v
A0v(t) zv= r0(0, t, z). (2.12)
Remark 2.8. Formula (2.10) implies that A1v(t), like Av(t), does not depend on the orientation of the
quiver Q. This is not the case for A0v(t) in general. However, since indecomposability is obviously preserved under transposition, the polynomial A0
v(t) is invariant under changing the direction of all
2.3. Proof of Proposition 1.2.
To finish the proof of Proposition 1.2 we must prove that for each ♭ = 0, 1 we have A♭v(t) ∈ Z[t], A♭v(1) = Av(1).
2.3.1. The integrality. We will use the following argument due to Katz, see [12, §6].
Lemma 2.9(Katz). Let Z be a constructible set defined over Fq. Assume that there exists a polynomial
P (t) in C[t] such that for any integer l > 1 we have |Z(Fql)| = P (ql). Then P (t) ∈ Z[t]. ⊓⊔
In order to use the above result, we must relate the polynomials A0
v(t) and A1v(t) to the point count
of some constructible sets. Set k = Fq. The sets of nilpotent and 1-nilpotent representations in Ev is
the set of Fq-points of the Zariski closed subvarieties Ev0 and Ev1. For ♭ = 0 or 1 we put
Aut♭v(Fq) = {(x, g) ∈ Ev♭(Fq) × GLv(Fq) ; g ∈ AutRepFq(x)},
Aut♭, a.i.v (Fq) = {(x, g) ∈ Aut♭v(Fq) ; x is absolutely indecomposable}.
Then Aut♭v(Fq) is the set of Fq-points of an algebraic variety defined over Fq, while Aut♭, a.i.v (Fq) is a
constructible subset of Autv(Fq). The set Aut♭,a.i.v (Fq) is compatible with field extensions, i.e., we have
Aut♭, a.i.v (Fql) = {(x, g) ∈ Ev♭(Fql) × GLv(Fql) ; x is absolutely indecomposable , g ∈ AutRepF ql(x)}.
This property is not true if one replaces absolutely indecomposable by indecomposable. Hence, for any integer l > 1, we have
|Aut♭, a.i. v (Fql)|
|GLv(Fql)|
= |{x ∈ Ev♭(Fql) ; x is absolutely indecomposable}/ ∼ | = A♭v(ql).
By Lemma 2.9 we deduce that
A♭v(t) ·Y
i vYi−1
k=0
(tvi− tk) ∈ Z[t].
From the facts that A♭v(t) ∈ Q[t] and
Q
i
Qvi−1
k=0(tvi − tk) is monic we deduce that A♭v(t) ∈ Z[t] as
wanted.
2.3.2. The value at 1. Let us now prove the equality A1v(1) = Av(1). Denote by Indv(Fq) and Ind1v(Fq)
the set of isomorphism classes of absolutely indecomposable and 1-nilpotent absolutely indecomposable representations of Q of dimension v over Fq. Let the finite group F×q act on Indv(Fq) as follows :
u · x = (uǫ(h)xh)h with ǫ(h) =
(
1 if h′ 6= h′′
u if h′ = h.
Let us consider the possible sizes of the F×q-orbits in Indv(Fq). If x ∈ Indv(Fq) and u ∈ F×q are such
that u · x ≃ x then there is an element g = (gi) ∈ GLv(Fq) such that we have u · xg = gx. In particular,
if h is a 1-loop at the vertex i then xh maps the generalized eigenspace Vi,λ of gi to Vi,uλ. We deduce
is a non-nilpotent polynomial P (xh1, . . . , xhgi) in the 1-loops h1, . . . , hgi at i then there is an integer
l ∈ (0, vi] such that uldeg(P )= 1.
Lemma 2.10. Let k be a field, V a finite-dimensional k-vector space of dimension d and {x1, . . . , xn} ⊂
End(V ). If all the polynomials P (x1, . . . , xn) with no constant terms of degree at most d2 are nilpotent
then there exists a flag of subspaces W• in V such that xi(Wj) ⊆ Wj−1 for all i, j.
Proof. By Engel’s theorem, it is enough to prove that the Lie subalgebra of gl(V ) generated by {x1, . . . , xn} consists of nilpotent elements. Set U = Pikxi. Consider the increasing filtration
U = U0 ⊆ U1 ⊆ · · · , where Ui = Ui−1+ [U, Ui−1]. This filtration stabilizes at, say, Uk to the Lie
subalgebra generated by {x1, . . . , xn}. Moreover Ul 6= Ul+1 for all l < k. Hence k 6 d2. It remains to
notice that any element of Uk is a polynomial in x1, . . . , xn of degree at most k. ⊓⊔
We deduce from Lemma 2.10 that an element x ∈ Indv(Fq) is not 1-nilpotent if and only if there
exists some vertex i and some polynomial P (xh1, . . . , xhgi) with zero constant term of degree t 6 v
2 i
and consisting of 1-loops at i, which is not nilpotent. Set N = sup{vi3; i ∈ I}. By the above, any u ∈ F×
q stabilizing a non 1-nilpotent element x ∈ Indv(Fq), satisfies uk= 1 for some integer k ∈ (0, N ].
Hence for any x ∈ Indv(Fq)\Ind0v(Fq) we have
| StabF× q(x)| 6 N X k=1 |µk(Fq)| 6 N 2 . It follows that for such x we have
|F×q · x| ∈ q − 1N
2
!N. Thus, for any field Fq we have
(q − 1)−1(Av(q) − A1v(q)) = (q − 1)−1· |Indv(Fq)\Ind0v(Fq)| ∈ 1 N 2 !N. (2.13)
This implies that the polynomial Av(t) − A1v(t) is divisible by (t − 1). Otherwise, we would have
1 t − 1(Av(t) − A 1 v(t)) ∈ Z[t] + c t − 1 for some nonzero c ∈ Z and (2.13) would be false for q ≫ 1.
The proof that A0
v(1) = Av(1) follows the same lines. Denote by Ind0v(Fq) the set of all absolutely
indecomposable nilpotent representations of Q of dimension v. We now use the action of F×q on Indv(Fq)
and Ind0v(Fq) defined by
u · x = u · (xh) = (uxh).
Arguing as above, we see that the stabilizer of any representation x ∈ Indv(Fq)\Ind0v(Fq) consists of
elements u ∈ F×q satisfying uk= 1 for some k ∈ (0, N′], where N′ is an integer depending on v but not on q. Specifically, one can take N′ = (Pivi)3. The rest of the argument is the same.
Example 2.11. Let us consider the case of the quiver with one vertex and g loops. Then A11(t) = 1, A12(t) = t g− 1 t − 1 , A13(t) = t2(g−1)+t 2(g−1)− 1 t2− 1 · tg+1− 1 t − 1 + tg− 1 t − 1 whereas A1(t) = tg, A2(t) = t2g−1 t2g− 1 t2− 1, A3(t) = t9g−3− t5g+1− t5g− t5g−1+ t3g−1+ t3g−2 (t2− 1)(t3− 1) .
Further, we have A1(1) = A11(1) = 1, A2(1) = A21(1) = g and A3(1) = A13(1) = 2g2− g.
3. Nakajima’s quiver varieties and their attracting subvarieties
In this section we consider variants of Λv and Λ♭v in the context of Nakajima quiver varieties, and
compute their number of points over finite fields. This will then be used in the next section to prove Theorem 1.4. Let k be any field. When we want to specify the field over which we consider a variety X we write X/k.
3.1. Quiver varieties.
Let Q be a finite quiver. We recall the definition of Nakajima quiver varieties and state some of their properties. See [28] for details, and see [24, §4] in the case of an arbitrary field. Fix v, w ∈ NI. Let V, W be I-graded k-vector spaces of dimensions v, w and set
M (v, w) =M h∈ ¯Ω Hom(Vh′, V h′′) ⊕ M i∈I Hom(Vi, Wi) ⊕ M i∈I Hom(Wi, Vi).
Elements of M (v, w) will be denoted m = (¯x, p, q). The space M (v, w) carries a natural symplectic structure, and the group Gv =QiGL(Vi) acts in a Hamiltonian fashion. The moment map is
µ : M (v, w) −→M i gl(Vi), (¯x, p, q) 7→ X h∈Ω h′ =i xh∗x h− X h∈Ω h′′ =i xhxh∗+ q ipi; i ∈ I (3.1)
The categorical quotient of µ−1(0) by G
v is the k-variety M0(v, w) = µ−1(0)//Gv = Spec M n∈N k[µ−1(0)]Gv .
It is affine and singular in general. Let [¯x, p, q] denote the image in M0(v, w) of the triple m = (¯x, p, q)
M0(v, w) is to the set of closed Gv(k)-orbits in µ−1(0)(k) so that [¯x, p, q] is the unique closed orbit in
the closure of the Gv(k)-orbit of (¯x, p, q).
Let θ be the character of Gv defined by θ(gi) = Qi∈Idet(gi)−1. Consider the smooth symplectic
quasiprojective variety of dimension 2d(v, w) = 2v · w − (v, v) given by M(v, w) = Proj M
n∈N
k[µ−1(0)]θn,
where
k[µ−1(0)]θn = {f ∈ k[µ−1(0)] ; f (g · m) = θn(g)f (m) , ∀m ∈ µ−1(0)}. There is a natural projective morphism
π : M(v, w) → M0(v, w).
We say that an element (¯x, p, q) ∈ µ−1(0)(k) is semistable if the following condition is satisfied : V′ ⊂M
i
Ker(pi) and ¯x(V′) ⊂ V′ ⇒ V′ = {0}.
Let µ−1(0)s be the open subset of µ−1(0) consisting of semistable points. Using the Hilbert-Mumford criterion one finds that if k is algebraically closed then M(v, w) is the geometric quotient
M(v, w) = µ−1(0)s//Gv
and the map π takes the Gv-orbit of (¯x, p, q) to [¯x, p, q].
Consider the following closed subvarieties of M(v, w) :
L0(v, w) = {Gv· (¯x, p, q) ; q = 0, ¯x is semi-nilpotent}, L1(v, w) = {Gv· (¯x, p, q) ; q = 0, ¯x is strongly semi-nilpotent}, L(v, w) = {Gv· (¯x, p, q) ; q = 0, ¯x is nilpotent}, M0(v, w) = {Gv· (¯x, p, q) ∈ M(v, w) ; x is nilpotent}, M1(v, w) = {Gv· (¯x, p, q) ∈ M(v, w) ; x is 1-nilpotent}. By definition, we have L0(v, w) =hµ−1(0)s∩Λ0v×M i Hom(Vi, Wi) i. Gv,
and a similar description L1(v, w), L(v, w) and M♭(v, w). Note that the definitions of M0(v, w) and M1(v, w) only involve the edges in Ω. The variety L(v, w) = π−1(0) is projective. If Q has no 1-loop then we have
L(v, w) = L1(v, w), M(v, w) = M1(v, w). If Q has no oriented cycle besides products of 1-loops then we have
L1(v, w) = L0(v, w), M0(v, w) = M1(v, w). In general the following inclusions are strict
The variety L(v, w) may not be equidimensional. The variety L1(v, w) is Lagrangian in M(v, w)
by [2, thm 1.15]. The same holds for L0(v, w). Their irreducible components may be parametrized by connected components of the variety of fixed points in M(v, w) under suitable torus actions, see Proposition 3.5 below.
Replacing Q by its opposite Q∗ we define the subvarieties L∗,0(v, w) and M∗,0(v, w) of M∗(v, w). Note that M∗(v, w) and M(v, w) are canonically isomorphic. Under this isomorphism we have L0(v, w) = L∗,0(v, w). We have the following chain of inclusions
L1(v, w) //L0(v, w) //M0(v, w) //M1(v, w) & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ L(v, w) & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ 8 8 q q q q q q q q q q M(v, w) L∗,1(v, w) //L∗,0(v, w) //M∗,0(v, w) //M∗,1(v, w) 8 8 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣
3.2. Reminder on tori actions on varieties.
3.2.1. The Bialynicki-Birula decomposition. An affine space bundle of rank d over X is a map f : Y → X such that X can be covered by open affine subsets Uisuch that f−1(Ui) ≃ Ad×Uiand f corresponds
under this bijection to the projection on the second factor.
The proof of Bialynicki-Birula uses the assumption that the field k is algebraically closed (of any characteristic) and that the variety X is smooth and projective. We’ll need it for any field k and for non smooth quasi-projective varieties X. The restriction on the field is not essential, since given a partition X =FρXρ of a Fq-variety X into locally closed subsets with affine space bundles pρ : Xρ → Fρ we
can always assume that the decomposition and the maps pρ are defined over Fq up to taking the field
Fq large enough to contain the field of definition of X, Xρ, Fρand pρ for all parameter ρ.
In the general situation, it is possible to use Hesselink’s generalization of the Bialynicki-Birula decomposition [14, thm 5.8] which holds true for arbitrary fields k and for non smooth quasi-projective varieties X. In particular, the following holds. Let X be an Fq-variety with a Gm-action. Assume that
X embeds equivariantly in a projective space with a diagonalizable Gm-action, or, equivalently, that
we have a very ample Gm-linearized line bundle over X. Assume also that the Gm-fixed point locus
XGm is contained in the regular locus of X. Define the attracting variety X′ as follows :
X′ = {x ∈ X ; lim
t→0t · x exists}.
Then we have a partition X′ =FρXρ into locally closed subsets with affine space bundles pρ as above
such that XGm =F
ρFρ is the decomposition into connected components. Further, there is a filtration
(X6ρ) of X by closed subsets and an ordering of the components Fρsuch that Xρ= X6ρ\ X<ρ. Note
that this filtration depends on the choice of the equivariant embedding of X in a projective space with a diagonalizable Gm-action.
3.2.2. Roots. Let T be the n-dimensional torus (Gm)n. Let
X∗(T ) = Hom(Gm, T ), X∗(T ) = Hom(T, Gm)
be the groups of cocharacters and characters of T . We identify both with Zn in the obvious way. The set of roots of a smooth T -variety X is the set of characters of T appearing in the normal bundles to the connected components of the fixed points locus XT. We’ll say that a cocharacter γ ∈ X∗(T ) is
generic if the canonical pairing α · γ is nonzero for each root α ∈ X∗(T ). In this case, the fixed point
locus Xγ= Xγ(Gm) coincides with XT.
3.3. The Bialynicki-Birula decompositions of M(v, w).
The subvarieties L♭(v, w), L(v, w) and M♭(v, w) of M(v, w) all have a geometric origin given in
terms of attracting subvarieties for suitable Gm-actions on M(v, w). Let us explain this.
3.3.1. The cases of L0(v, w), L(v, w) and M0(v, w). Let T = (Gm)2 be the two-dimensional torus
acting on the k-variety M (v, w) by
(t1, t2) · (¯x, p, q) = (t1x, t2x∗, t2p, t1q).
This yields a T -action on M(v, w), M0(v, w) such that the map π is T -equivariant and
M0(v, w)T = {0}.
We deduce that the fixed point variety M(v, w)T decomposes as a disjoint union M(v, w)T = G
ρ∈χ
Fρ
of smooth connected projective varieties Fρ which embed into L(v, w).
Now, given a cocharacter γ, the variety M(v, w)γ is smooth, not necessarily projective and it is the
disjoint union of smooth connected closed subvarieties M(v, w)γ = G
κ∈χγ
Fκγ. Let Mγ ⊆ M(v, w) be the attracting subvariety defined by
Mγ = {x ∈ M(v, w) ; lim
t→0γ(t) · x exists}.
Proposition 3.1. Let γ = (a, b) and fix v, w in NI. If the field k is large enough then (a) a, b > 0 ⇒ Mγ = M(v, w), (b) a, b < 0 ⇒ Mγ = L(v, w), (c) a < 0 = b ⇒ Mγ= L0(v, w), b < 0 = a ⇒ Mγ= L∗,0(v, w), (d) a < 0 < b with |b/a| ≫ 1 ⇒ Mγ= M0(v, w), b < 0 < a with |a/b| ≫ 1 ⇒ Mγ= M∗,0(v, w).
Proof. Parts (a) and (b), are well-known, see [27]. Part (c) is stated in [2, §2.1]. Part (d) is proved using a similar argument. For the convenience of the reader, we give a proof of both here. Since we may assume that the field k is as large as we want, we may as well assume that it is indeed algebraically closed. In addition, it is clearly enough to treat the first statements of (c) and (d).
We have
L0(v, w) = π−1π(L0(v, w)), M0(v, w) = π−1π(M0(v, w)).
To see this, we may for instance use the description of the projective map π : M(v, w) → M0(v, w)
as a semi-simplification map, see [28, prop. 3.20] together with the fact that a filtered representation x ∈ Ev is semi-nilpotent if and only its associated graded is (resp. a filtered representation x ∈ Ev is
nilpotent if and only if its associated graded is).
Since the map π is projective and T -equivariant, for any value of γ we have Mγ= π−1π(Mγ), π(Mγ) = {x ∈ M0(v, w) ; lim
t→0γ(t) · x exists}.
Hence it suffices to prove that for a < 0 = b we have
π(L0(v, w)) = {x ∈ M0(v, w) ; lim
t→0γ(t) · x exists} (3.2)
while for a < 0 < b and |b/a| ≫ 1 we have
π(M0(v, w)) = {x ∈ M0(v, w) ; lim
t→0γ(t) · x exists} (3.3)
For each path σ in ¯Q and each tuple (¯x, p, q) ∈ µ−1(0), let ¯xσ be the composition of ¯x along σ. We’ll
view it as an element of
¯
xσ ∈ k[µ−1(0)] ⊗ End(V ).
By [22, thm. 1.3], [19] if char(k) = 0, and [7] along with Crawley-Boevey’s trick in [5] if char(k) = p > 0, the algebra k[M0(v, w)] is generated by the following two families of functions :
(Aσ) : the coefficients of the characteristic polynomials of ¯xσ for each oriented cycles σ in ¯Q,
(Bσ) : the functions φ(p ¯xσq) for each linear form φ on End(W ) and each path σ in ¯Q.
The closed subset π(Mγ) of M0(v, w) is the intersection of π(M(v, w)) with the zero set of all the
functions in (Aσ) or (Bσ) which are of negative weight under the γ-action. Our aim is to determine
precisely these sets under the assumption (c) (resp. (d)) on γ.
Let us first consider the situation in which γ = (a, b) is as in case (c). Let z = (¯x, p, q) ∈ µ−1(0)s
such that [z] ∈ Mγ. We first claim that q = 0. Indeed, otherwise, by the stability condition we have
p ¯xσq 6= 0 for some path σ in ¯Q. But this defines a regular function on M0(v, w) which is of negative
weight. Next, let us prove that ¯x ∈ Λ0v. By our choice of γ, the characteristic polynomial of ¯xσ for any
cycle in ¯Q containing at least one arrow in Q must be zero, and thus ¯xσ is nilpotent for any such cycle
σ. In particular, xσ is nilpotent for any cycle σ in Q. Let A denote the image of the path algebra k ¯Q
under the natural evaluation map σ 7→ ¯xσ, and let A+⊂ A denote the unital subalgebra generated by
all the paths containing an arrow from Q. By the above, A+ is a finite-dimensional algebra consisting of nilpotent endomorphisms hence by Wedderburn’s theorem, A+ is nilpotent. But then the flag of I-graded subspaces K• defined by Kl= Im((A+)l) satisfies V = K0 ⊃ K1⊃ · · · ⊃ Kn= {0} for some
large enough n, and
Therefore ¯x is indeed semi-nilpotent, and z ∈ µ−1(0)s∩ (Λ0 v×
L
iHom(Vi, Wi)). We have shown that
Mγ ⊆ L0(v, w). To prove the reverse inclusion, it is enough to notice that if z ∈ µ−1(0)s∩ (Λ0v ×
L
iHom(Vi, Wi)) then by the same argument as above, all the regular functions of negative weight on
M0(v, w) vanish on π([z]), and thus [z] ∈ Mγ.
The argument in case (d) is very similar. Since a < 0, for any oriented cycle σ in Q the characteristic polynomial of the monomial xσ has a negative weight. Hence, it vanishes on π(Mγ). As above, we
deduce that if [¯x, p, q] is a k-point of π(Mγ) then xσ is nilpotent for all such σ, hence x is nilpotent.
Therefore, we have
π(Mγ) ⊆ π(M0(v, w)).
It remains to prove the reverse inclusion if b is large enough. Let m = (¯x, p, q) such that [m] belongs to M0(v, w), i.e., the representation x is nilpotent. We must prove that any function of type (A
σ) or
(Bσ) which is of negative weight under the γ-action vanishes at m. We will prove this for functions of
type (Aσ) and leave the other case to the reader.
Let σ be an oriented cycle in ¯Q. The weight of ¯xσ under the γ-action is of the form
a |σ ∩ Q| + b |σ ∩ Q∗|.
Assume that it is negative. Assume also that b > 2N |a| for some large enough positive integer N . Then, we have
|σ ∩ Q| > 2N |σ ∩ Q∗|.
If σ ∩ Q∗ = ∅, then σ is a path in Q, hence the characteristic polynomial of ¯xσ vanishes because the
representation x is nilpotent. If σ ∩ Q∗ 6= ∅, then there is at least N consecutive arrows in σ ∩ Q. Hence ¯
xσ vanishes because the representation x is nilpotent. ⊓⊔
3.3.2. The cases of L1(v, w) and M1(v, w). This is very similar to the above, using a different torus action. The torus T′ = {(t
1, t2, t3, t4) ∈ G4m; t1t2 = t3t4} linearly acts on ¯Ev as follows :
(t1, t2, t3, t4) · xh = t1xh if h′ 6= h′′, h ∈ Ω t2xh if h′ 6= h′′, h ∈ Ω∗ t3xh if h′ = h′′, h ∈ Ω t4xh if h′ = h′′, h ∈ Ω∗.
Consider the action on M(v, w) defined as
(t1, t2, t3, t4) · (¯x, p, q) = ((t1, t2, t3, t4) · ¯x , p , t1t2q)
The group of cocharacters of T′ is identified with {(u1, u2, u3, u4) ∈ Z4 ; u1+ u2 = u3+ u4}. For any
cocharacter γ let Mγ be the attracting variety of the corresponding Gm-action on M(v, w).
The following can be proved by the same type of arguments as in Proposition 3.1. Proposition 3.2. Fix v, w ∈ NI and γ = (u
1, u2, u3, u4) ∈ X∗(T′). If the field k is large enough, then
(a) u1, u2, u3 < 0, u4 = 0 ⇒ Mγ = L1(v, w),
(b) u1, u2, u3 > 0, u4 < 0 and |u1/u4|, |u2/u4|, |u3/u4| ≫ 1 ⇒ Mγ = M∗,1(v, w).
3.4. More on the Bialynicki-Birula decompositions of M(v, w).
We now draw some consequences of the Bialynicki-Birula decompositions considered in the previous sections. Again, we work out everything in details for L0(v, w), L(v, w) and state the analogous results for L1(v, w).
3.4.1. The cases of L0(v, w), L(v, w). Fix v, w in NI and fix a cocharacter γ = (a, b) of T . Assume
that either the field k is algebraically closed or that it is finite with a large enough number of elements. Consider the Gm-action on the k-variety Mγ associated with the cocharacter γ. The Bialynicki-Birula
decomposition gives a partition
Mγ =
G
κ∈χγ
Mγ,κ
into locally closed k-subvarieties Mγ,κ and affine space bundles pγ,κ : Mγ,κ→ Fγ,κ such that
M(v, w)γ = G
κ∈χγ
Fγ,κ
is the decomposition of the γ-fixed point set of M(v, w) into a disjoint union of smooth connected closed subvarieties.
In order to relate the dimension of the affine space bundle pγ,κ for various γ and κ, we consider the
restriction of pγ,κ to the set of T -fixed points. Each Fγ,κ is T -stable. We deduce that we have
G ρ∈χ Fρ= M(v, w)T = G κ∈χγ (Fγ,κ)T.
This provides us with a map πγ : χ → χγ such that we have
(Fγ,κ)T =
G
ρ∈π−1
γ (κ)
Fρ.
We claim that the map πγ is surjective. Indeed, by Proposition 3.1(a) there exists a generic cocharacter
σ ∈ X∗(T ) which acts on M(v, w) in a contracting way. Therefore, every T -orbit in M(v, w) contains
a σ-fixed point in its closure, and, by genericity of σ, a σ-fixed point is indeed fixed by T . Since each Fγ,κ is closed and T -invariant, it follows that each Fγ,κ contains a T -fixed point.
Let ∆(v, w) be the set of roots of the T -action on M(v, w). It is a finite set. Pick a point zρ∈ Fρ
for each ρ. Decompose the tangent space Tρ of M(v, w) at zρ as a sum of T -weight spaces. We have
Tρ=
M
α∈∆(v,w)
Tρ[α], (3.4)
where the torus T acts on Tρ[α] via the character α. The multiplicity dim(Tρ[α]) does not depend on
the choice of the element zρ in Fρ, because Fρis connected.
Let ∆(v, w)γ, ∆+(v, w)γ and ∆−(v, w)γ be the set of roots in ∆(v, w) given by
For any κ ∈ χγ and ρ ∈ χ such that πγ(ρ) = κ we have dim(Fγ,κ) = X α∈∆(v,w)γ dim(Tρ[α]), dim(pκ,γ) = X α∈∆+(v,w)γ dim(Tρ[α]). (3.5)
Since the T -action on M(v, w) scales the symplectic form ω by the character ω := (1, 1), the form ω restricts to a nondegenerate bilinear form
ω : Tρ[α] × Tρ[ω − α] → k.
In particular, we have the following formula for each ρ and α
dim(Tρ[α]) = dim(Tρ[ω − α]). (3.6)
Recall that the cocharacter γ is generic if the set ∆(v, w)γ is empty. If γ is generic, then we have
M(v, w)γ= M(v, w)T and the map πγ is a bijection χ → χγ such that Fγ,πγ(ρ) = Fρ.
Let us now examine the equation (3.5) in each of the cases occuring in Proposition 3.1. (a) Choose a, b > 0 generic. Then for any ρ ∈ χγ= χ we have
Mγ = M(v, w), M−γ = L(v, w),
dim(pγ,ρ) + dim(p−γ,ρ) + dim(Fρ) = dim(M(v, w)).
(3.7) (b) Choose a < 0 = b. Thus γ may not be generic. Then, for any κ ∈ χγ and ρ ∈ π−1γ (κ) we have
Mγ = L0(v, w), dim(pγ,κ) = X k<0 X l∈Z dim(Tρ[k, l]). (3.8)
Dually, if b < 0 = a, then for any κ ∈ χγ and ρ ∈ π−1γ (κ) we have
Mγ = L∗,0(v, w), dim(pγ,κ) = X k∈Z X l<0 dim(Tρ[k, l]). (3.9)
(c) Choose a < 0 < b generic with |b|/|a| ≫ 1. Then for any ρ ∈ χγ = χ we have
Mγ = M0(v, w), dim(pγ,ρ) = X k∈Z X l>0 dim(Tρ[k, l]) + X k<0 dim(Tρ[k, 0]). (3.10)
Dually, if a > 0 > b generic with |a|/|b| ≫ 1 then we have Mγ = M∗,0(v, w), dim(pγ,ρ) = X k>0 X l∈Z dim(Tρ[k, l]) + X l<0 dim(Tρ[0, l]). (3.11)
By an almost affine space bundle we’ll mean a map which is a composition of affine space bundles. We can now prove the following.
(a) There exists partitions into locally closed k-subvarieties M(v, w) = G ρ∈χ Zρ, L(v, w) = G ρ∈χ Yρ
and affine space bundles uρ: Zρ→ Fρ and vρ: Yρ→ Fρ such that
dim(uρ) + dim(vρ) + dim(Fρ) = dim(M(v, w)). (3.12)
(b) There exists partitions into locally closed k-subvarieties M∗,0(v, w) = G ρ∈χ Zρ, L0(v, w) = G ρ∈χ Yρ
and almost affine space bundles uρ: Zρ→ Fρ and vρ: Yρ→ Fρ such that
dim(uρ) + dim(vρ) + dim(Fρ) = dim(M(v, w)). (3.13)
Proof. To prove the claim (a), we choose a, b as in (3.7) and set
Zρ= Mγ, ρ, uρ= pγ, ρ, Yρ= M−γ, ρ, vρ= p−γ, ρ.
Let us turn to the claim (b). Given a, b as in (3.10), we put Zρ= Mγ, ρ, uρ= pγ, ρ.
To construct the partition of L0(v, w) we proceed in two steps. For γ = (−1, 0) we consider the partition
L0(v, w) =G
κ
Mγ,κ
and the affine space bundles pγ,κ : Mγ,κ → Fγ,κ. Now pick some generic a′, b′ > 0. The γ′-action on
M(v, w) is contracting. Since Fγ,κ is smooth and T -stable, the Bialynicki-Birula decomposition gives
us with a partition Fγ,κ= G ρ∈π−1 γ (κ) Uρ,κ
and affine space bundles qρ,κ : Uρ,κ → Fρ such that we have
dim(qρ,κ) = X α∈∆(v,w)γ∩∆+(v,w)γ′ dim(Tρ[α]) = X l>0 dim(Tρ[0, l]). (3.14)
Now, for each ρ ∈ χ we set κ = πγ(ρ) and we define
Using (3.8), (3.11) and (3.14) we compute
dim(uρ) + dim(vρ) + dim(Fρ) = dim(uρ) + dim(pγ,κ) + dim(qρ,κ) + dim(Fρ)
=X k>0 X l∈Z dim(Tρ[k, l]) + X l<0 dim(Tρ[0, l]) + X k<0 X l∈Z dim(Tρ[k, l])+ +X l>0 dim(Tρ[0, l]) + dim(Tρ[0, 0]) = dim(Tρ) = dim(M(v, w)) as wanted. ⊓⊔
3.4.2. The cases of L1(v, w). Fix again some v, w and assume that the ground field k is algebraically closed or that it is finite with a large enough number of elements. The same arguments as above, applied to the decompositions resulting from the T′-actions described in §3.3.2 yields the following result. Let us denote by
M(v, w)T′ = G
ρ∈χ′
Fρ
the decomposition of the T′-fixed point subvariety of M(v, w) into connected components. Each Fρis
smooth and projective as M(v, w)T′
⊂ π−1(0).
Proposition 3.4. There exists partitions into locally closed k-subvarieties M∗,1(v, w) = G ρ∈χ′ Zρ, L1(v, w) = G ρ∈χ′ Yρ
and affine space bundles uρ: Zρ→ Fρ and vρ: Yρ→ Fρ such that
dim(uρ) + dim(vρ) + dim(Fρ) = dim(M(v, w)). (3.15)
3.4.3. Irreducible components of L0(v, w) and L1(v, w). The following simple corollary of (3.6) pro-vides a geometric description and parametrization of the irreducible components of L0(v, w) and
L1(v, w).
Proposition 3.5.
(a) The variety L0(v, w) is Lagrangian in M(v, w). Fix γ = (a, b) with a < 0 = b. For any κ ∈ χ γ,
the smooth variety Mγ,κ is pure dimensional, and
dim(Mγ,κ) = dim(L0(v, w)) =
1
2dim(M(v, w)).
In particular, the irreducible components of L0(v, w) are precisely the (Zariski closures) of the inverse images under the affine space bundles pγ,κ of the connected components of Fγ,κ for κ ∈ χγ.
(b) The variety L1(v, w) is Lagrangian in M(v, w) and is a closed subvariety in L0(v, w). In partic-ular, the irreducible components of L1(v, w) are the (Zariski closures) of the inverse images under
Proof. The argument is similar to [28, thm. 5.8]. To prove (a) it is enough to show that for any κ ∈ χγ
we have
dim(pγ,κ) + dim(Fγ,κ) =
1
2dim(M(v, w)). Fix κ ∈ χγ, and zρ∈ Fρ⊂ Fγ,κ for some ρ ∈ πγ−1(κ) . By (3.8) we have
dim(pγ,κ) = X k<0,l∈Z dim(Tρ[k, l]), dim(Fγ,κ) = X l∈Z dim(Tρ[0, l]). But by (3.6) we have X k≤0,l∈Z dim(Tρ[k, l]) = X k>0,l∈Z dim(Tρ[k, l]) = 1 2 X k,l dim(Tρ[k, l]) = 1 2dim(M(v, w))
from which we deduce the desired equality of dimension. The proof that L1(v, w) is Lagrangian can
be found in [1]. The other statements follow from (a). ⊓⊔
3.5. Counting polynomials of quiver varieties.
In this paragraph we fix v, w and assume that the field Fq is large enough to contain the field of
definition of the quiver varieties M(v, w), L(v, w), etc, and the field of definition of the partitions and the affine space bundles considered in the previous section.
We will say that an Fq-algebraic variety X has polynomial count if there exists a polynomial P (X, t)
in Q[t] such that for any finite extension k of Fq we have |X(k)| = P (X, |k|).
Let ℓ 6= p be a prime number. We will say that an Fq-algebraic variety X is ℓ-pure (resp. very ℓ-pure)
if all eigenvalues of the Frobenius endomorphims F of the ℓ-adic cohomology groups Hci(X ⊗ Fq, Qℓ)
are of norm qi/2 (resp. are equal to qi/2). The Grothendieck-Lefschetz formula implies that |X(Fqr)| =
X
i
(−1)iTr(Fr, Hci(X ⊗ Fq, Qℓ)).
Hence if X is very ℓ-pure then it has polynomial count and its counting polynomial coincides with its ℓ-adic Poincar´e polynomial
Pc(X, t) :=
X
i
dim Hc2i(X ⊗ Fq, Qℓ) ti
Conversely, it is a classical result that if X is of polynomial count and ℓ-pure then it is very ℓ-pure and has no odd cohomology see, e.g., [6, lem. A.1]. Further, the counting polynomial P (X, t) belongs to N[t].
The following result may be found in [11, thm. 1], see also [24, prop. 6.1, thm.6.3].
Theorem 3.6 (Hausel). The variety M(v, w) is very ℓ-pure and of polynomial count. Its counting polynomial (which is equal to its Poincar´e polynomial in ℓ-adic cohomology) is determined by the equality
X
v
t−d(v,w)P (M(v, w), t) zv= r(w, t, z)
